For nonintegrable weight , some weight multiplicities of the irreducible module over affine Lie algebras are expressed in terms of the colored partition functions. Also we find the multiplicity of in ther Verma module for any affine Lie algebras.

In this paper, we prove that $x^{1+\frac{q-1}{5}} + ax (a \neq 0)$ is not a permutation polynomial over $F_{q^r} (r \geq 2)$ and we show some properties of $x^{1+\frac{q-1}{m}} + ax (a \neq 0)$ over $F_{q^r} (r \geq 2)$.

In this paper, we introduce a permutation graph over a graph G as a generalization of both a graph bundle over G and a standard permutation graph, and study a characterization of a natural isomorphism and an automorphism of permutation graphs over a graph.

For the given torsion theory , we study some equivalent conditions when the localized ring be semisimple artinian (Theorem 4). Using this, if is semisimple artinian ring, we study when does the given ring R become left V-ring\ulcorner

The interpolation of scattered data with radial basis functions is knwon for its good fitting. But if data get large, the coefficient matrix becomes almost singular. We introduce different knots and nodes to improve condition number of coefficient matrix. The singulaity of new coefficient matrix is investigated here.

Linear invariance is closely related to the concept of uniform local univalence. We give a geometric proof that a holomorphic locally univalent function defined on the open unit disk is linearly invariant if and only if it is uniformly locally univalent.

Let denote the class of the Spirallike functions of order $\alpha, 0 < \alpha < \frac{\pi}{2}$ Let $\Pi_N$ denote the subset of consisting of all products where $m = 1 + e^{-2i\alpha},u_j = 1, t_j > 0$ for $j = 1, and may be found which lie in for some . We are let to conjecture that all exreme points of $S_p(\alpha)$ lie in for somer and that every such function is an extreme point.

We discuss certain structure theorems in the class A which is closely related to the study of the problems of solving systems concerning the predual of a dual operator algebra generated by a contraction on a separable infinite dimensional complex Hilbert space.

The unilateral weighted shift operator with the weighted sequence is unicellular if $0 < r < 1$. In general, A + B is not unicellular even if A and B are unicellular. We will prove that is unicellular if $0 < r < 1$.

Let $P_I$ be the convex compact set of all unital positive linear maps between the matrix algebra over the complex field. We find a necessary and sufficient condition for which two maximal faces of intersect. In particular, we show that any pair of maximal faces of has the nonempty intersection, whenever .

Consider the problem $-div(\bigtriangledown_u^{p-2}\bigtriangledown_u) = u^{p^*-2}u + \lambdau^{q-2}u$ in B, u = 0 on $\partial B$; where is a ball, $\lambda < 0, 1 < p < n$ and is the critical Sobolev exponent. For given $\lambda > 0$, we show that there exists such that any radial solutions to this problem have at most k noda curves when .

The definitions and techniques, which deals with homotopic harmonic maps from a compact Riemannian manifold into a compact metric space, developed by N. J. Korevaar and R. M. Schoen [7] can be applied to more general situations. In this paper, we prove that for a complicated domain, possibly noncompact Riemannian manifold with infinitely generated fundamental group, the existence of homotopic harmonic maps can be proved if the initial map is simple in some sense.

Assume that X is locally compact and Hausdorff. Then, we show that $\alpha X = sup {X \cup_f S(f)f \in S^{\alpha}}$ for any compactification of X if and only if for any 2-point compatification of X with , there exists a clopen subset A of such that and . As a corollary, we obtain that if X is connected and locally connected, then $\alpha X = sup {X \cup_f S(f)f \in S^{\alpha}}$ for any compactification of X if and only if X is 1-complemented.

Let f be a smooth G map from a nonsingular real algebraic G variety to an equivariant Grassmann variety. We use some G vector bundle theory to find a necessary and sufficient condition to approximate f by an entire rational G map. As an application we algebraically approximate a smooth G map between G spheres when G is an abelian group.

This paper first establishes some conditions for preconditioner under which PGCR does not break down. Next, VPGCR algorithm whose preconditioners can be easily obtained is introduced and then its breakdown and convergence properties are discussed. Lastly, implementation details of VPGCR are described and then numerical results of VPGCR with a certain criterion guaranteeing no breakdown are compared with those of restarted GMRES.